3,324 research outputs found
Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains
Existing approaches for multivariate functional principal component analysis
are restricted to data on the same one-dimensional interval. The presented
approach focuses on multivariate functional data on different domains that may
differ in dimension, e.g. functions and images. The theoretical basis for
multivariate functional principal component analysis is given in terms of a
Karhunen-Lo\`eve Theorem. For the practically relevant case of a finite
Karhunen-Lo\`eve representation, a relationship between univariate and
multivariate functional principal component analysis is established. This
offers an estimation strategy to calculate multivariate functional principal
components and scores based on their univariate counterparts. For the resulting
estimators, asymptotic results are derived. The approach can be extended to
finite univariate expansions in general, not necessarily orthonormal bases. It
is also applicable for sparse functional data or data with measurement error. A
flexible R-implementation is available on CRAN. The new method is shown to be
competitive to existing approaches for data observed on a common
one-dimensional domain. The motivating application is a neuroimaging study,
where the goal is to explore how longitudinal trajectories of a
neuropsychological test score covary with FDG-PET brain scans at baseline.
Supplementary material, including detailed proofs, additional simulation
results and software is available online.Comment: Revised Version. R-Code for the online appendix is available in the
.zip file associated with this article in subdirectory "/Software". The
software associated with this article is available on CRAN (packages funData
and MFPCA
Tree-valued Feller diffusion
We consider the evolution of the genealogy of the population currently alive
in a Feller branching diffusion model. In contrast to the approach via labeled
trees in the continuum random tree world, the genealogies are modeled as
equivalence classes of ultrametric measure spaces, the elements of the space
. This space is Polish and has a rich semigroup structure for the
genealogy. We focus on the evolution of the genealogy in time and the large
time asymptotics conditioned both on survival up to present time and on
survival forever. We prove existence, uniqueness and Feller property of
solutions of the martingale problem for this genealogy valued, i.e.,
-valued Feller diffusion. We give the precise relation to the
time-inhomogeneous -valued Fleming-Viot process. The uniqueness
is shown via Feynman-Kac duality with the distance matrix augmented Kingman
coalescent. Using a semigroup operation on , called concatenation,
together with the branching property we obtain a L{\'e}vy-Khintchine formula
for -valued Feller diffusion and we determine explicitly the
L{\'e}vy measure on . From this we obtain for
the decomposition into depth- subfamilies, a representation of the process
as concatenation of a Cox point process of genealogies of single ancestor
subfamilies. Furthermore, we will identify the -valued process
conditioned to survive until a finite time . We study long time asymptotics,
such as generalized quasi-equilibrium and Kolmogorov-Yaglom limit law on the
level of ultrametric measure spaces. We also obtain various representations of
the long time limits.Comment: 93 pages, replaced by revised versio
Generalized Functional Additive Mixed Models
We propose a comprehensive framework for additive regression models for
non-Gaussian functional responses, allowing for multiple (partially) nested or
crossed functional random effects with flexible correlation structures for,
e.g., spatial, temporal, or longitudinal functional data as well as linear and
nonlinear effects of functional and scalar covariates that may vary smoothly
over the index of the functional response. Our implementation handles
functional responses from any exponential family distribution as well as many
others like Beta- or scaled non-central -distributions. Development is
motivated by and evaluated on an application to large-scale longitudinal
feeding records of pigs. Results in extensive simulation studies as well as
replications of two previously published simulation studies for generalized
functional mixed models demonstrate the good performance of our proposal. The
approach is implemented in well-documented open source software in the "pffr()"
function in R-package "refund"
Restricted Likelihood Ratio Testing in Linear Mixed Models with General Error Covariance Structure
We consider the problem of testing for zero variance components in linear mixed models with correlated or heteroscedastic errors. In the case of independent and identically distributed errors, a valid test exists, which is based on the exact finite sample distribution of the restricted likelihood ratio test statistic under the null hypothesis. We propose to make use of a transformation to derive the (approximate) test distribution for the restricted likelihood ratio test statistic in the case of a general error covariance structure. The proposed test proves its value in simulations and is finally applied to an interesting question in the field of well-being economics
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